Smooth transition type integral operators
The paper investigates the integral operators of convolution type \((Ky)(x)=y(x)+\lambda (x)\int^{\infty}_{-\infty}k(x-t)y(t)dt\quad (y\in L_ 2(R)).\) Under the natural conditions on \(\lambda\) (x) and k(x) the connection of the operator k with integral operators on the semi-axis are found. In the particular case of \(\lambda (x)=t h(x/2+i\pi \beta /2)\) \((-1<\beta <1)\) from this follows the solution of the equation \(Ky=f\) which generalized the well-known Cherskij equation (corresponding to the case \(\lambda (x)=t h(x/2)).\) The matrix generalization of the operator of smooth transition is also solved. In the case of \(\lambda (x)=P(t h((x+i\pi \beta)/2))\) where P is a polynomial and \(\lambda (x)=\sum^{n}_{m=1}t h(x/2-\lambda_ m+i\pi \beta /2)\) \((\lambda_ m\in R)\) the investigation of the operator k reduces to the problem of factorization of certain matrices of functions.
scientific article; zbMATH DE number 4165689
scientific article